Firm-level return dispersion and the future volatility of aggregate stock market returns

Journal of Financial Markets 6 (2003) 389–411

Firm-level return dispersion and the future volatility of aggregate stock market returns$ Christopher T. Stivers* Terry College of Business, University of Georgia, Brooks Hall, Athens, GA 30602, USA

Abstract We find a sizeable positive relation between firm return dispersion and future market-level volatility in U.S. monthly equity returns from 1927 to 1995. This intertemporal relation remains strong when controlling for return shocks in the aggregate stock market, widely-used factor-mimicking portfolios, and government bonds. In contrast, the well-known positive relation between market-return shocks and future market-level volatility largely disappears when controlling for firm return dispersion. We also document how firm return dispersion moves with the contemporaneous market return and with economic conditions. Collectively, our evidence suggests that the time variation in firm return dispersion has important marketwide implications. r 2002 Elsevier Science B.V. All rights reserved. JEL classification: D80; G12; G14 Keywords: Equity market volatility; Return dispersion

$ This paper is based on part of my dissertation at the University of North Carolina at Chapel Hill. I am grateful for the assistance of my dissertation committee, Jennifer Conrad (chair), Dong-Hyun Ahn, Bob Connolly, Henri Servaes, and Steve Slezak. I also appreciate comments from Andrew Ang, Mike Cooper, Bruce Lehmann (the editor), Chris Mathews, Tod Perry, Lee Stivers, Pietro Veronesi, an anonymous referee, and seminar participants at the University of North Carolina, the University of Georgia, the University of Wisconsin-Milwaukee, and the 1997 Financial Management Association meetings. Additionally, I thank Eugene Fama for the three-factor portfolio return data. *Tel.: +1-706-542-3648. E-mail address: [email protected] (C.T. Stivers).

1386-4181/02/$ - see front matterr 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 4 1 8 1 ( 0 2 ) 0 0 0 4 4 - 7

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C.T. Stivers / Journal of Financial Markets 6 (2003) 389–411

1. Introduction This paper introduces new empirical evidence that further characterizes time variation in market-level volatility.1 Specifically, we investigate whether the dispersion in individual firm returns provides incremental information about future market-level volatility. By incremental, we mean volatility information beyond that provided by the well-known ARCH behavior in market returns. We use the crosssectional standard deviation of firm returns as a parsimonious measure of return dispersion (RD). Our motivation has two aspects. First, from a statistical perspective, firm RD may provide additional information to assist in identifying the unobservable market volatility in the period. With persistence in volatility, better identification of the current volatility should lead to a better estimate of future volatility. For example, firm RD might statistically capture information about a period’s multiple commonfactor shocks.2 Then, if the multiple common factors have variation in their volatility persistence, RD might provide incremental information about future market volatility. Second, from an economic perspective, both market-level volatility and firm RD are higher during recessions; and both positively co-vary with the yield spread between high and low rated corporate bonds (Schwert, 1989; Christie and Huang, 1994a). Further, Loungani et al. (1990) find that firm RD leads unemployment, which suggests that high RD is associated with transitional times of high economic reallocation across firms. These relations suggest that time variation in RD may tell us something about the economic state or economic transitions. If so, RD may provide incremental information about future market volatility. These statistical and economic perspectives are not mutually exclusive and both suggest that our empirical investigation may prove worthwhile. Our empirical testing focuses on the widely used asymmetric GARCH(1,1) framework in order to jointly model the conditional mean and volatility. We insert our RD variable in both the mean and variance GARCH equations. We study U.S. monthly equity returns over 1927–1995 and focus our analysis on the return and RD of large-firm portfolios, which are good proxies for the value-weighted market and are not subject to potential ‘small-firm return’ criticisms. We focus on the monthly horizon rather than the daily or weekly horizon for several reasons. First, relations in monthly returns may have more economic significance and are less likely to be driven by frictions or data measurement issues, especially for large-firm returns. Second, monthly firm data is available for approximately twice as long. Third, our priors are that any 1 See, e.g., Schwert (1989), Haugen et al. (1991), Bollerslev et al. (1992), Whitelaw (1994), and Campbell et al. (2001) for evidence of time-variation in stock market volatility. 2 We show that firm RD depends on the period’s market-return shock and firms’ residual variances from a market model process. These firm residual variances may reflect omitted common-factor shocks (Lehmann, 1990).

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economic information in firm return dispersion is likely to be noisier for the daily and weekly horizon. To anticipate our primary empirical findings, we find a sizable, reliable, and pervasive positive relation between RD and future market volatility. This intertemporal relation is a positive partial relation that remains strong while controlling for standard GARCH relations in the market return. Subperiod analysis yields consistent results. Further, we find a similar partial relation between firm RD and future market volatility while using other GARCH specifications, other statistical models, and alternate volatility measures. Next, we expand the conditional variance equation by adding other economic and market variables that are suggested by prior literature. We find that the incremental volatility information in RD remains strong when we control for the following: (1) the risk-free rate (Glosten et al., 1993); (2) the bond default yield spread (Schwert, 1989); (3) the recessionary state (Schwert, 1989); (4) the past volatility of the three factor-mimicking portfolios from Fama and French (1993); and (5) past government bond return volatility (Schwert, 1989; Scruggs, 1998). In contrast, we find that the positive relation between market-return shocks and future market volatility is much weaker when controlling for the risk-free rate, bond default yield spread, bond returns, and the recessionary state. Further, the marketreturn shock has no reliable positive relation to future market volatility after we control for firm RD, in any of our 15-year subperiods. Finally, we also find a large positive contemporaneous relation between RD and the absolute market return. This relation is much greater than the relation predicted by a market-model return process with constant market-betas and independently varying firm-specific volatility.3 Collectively, our contemporaneous and intertemporal findings suggest a common co-movement between market-level volatility, firmspecific volatility, and dispersion in factor loadings. The paper is organized as follows. Section 2 discusses firm return dispersion from the familiar perspective of a linear, common-factor, return process. We present the data and main empirical findings in Sections 3 and 4, respectively. Section 5 expands our analysis to control for economic conditions and other market variables; and Section 6 examines the contemporaneous relation between firm RD and the market-level return. Section 7 concludes and further discusses related literature.4

3

By firm-specific volatility, we mean firm volatility beyond the volatility that can be attributed to market movements in a single index market model. 4 Prior uses of firm RD in the literature include the following. Bekaert and Harvey (1997, 2000) use a firm RD measure and examine the relation between the level of a country’s market volatility and the level of the country’s firm return dispersion. For developed markets, they find that markets with higher RD tend to have a higher volatility level. In contrast, for emerging markets, they find that markets with a lower RD tend to have a higher volatility level. Second, Christie and Huang (1994b) and Chang et al. (2000) both use a firm RD measure to evaluate herding and find no evidence of herding in the U.S. Finally, Bessembinder et al. (1996) use firm RD as a measure of firm information flows and find that RD is correlated with aggregate firm-level volume.

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2. Firm return dispersion in a linear common-factor return framework 2.1. Overview In this section, we discuss firm return dispersion and time variation in market volatility from the familiar perspective of a linear, common-factor, return generating process (RGP). Section 2.2 discusses a simple null framework where RD would not provide any incremental information about future market-level volatility (information beyond that contained in market-return shocks). Sections 2.3 and 2.4 discuss alternate scenarios (but not necessarily mutually exclusive scenarios) where RD might provide incremental information about future market-level volatility. 2.2. RD in a single market-factor RGP with constant market-betas, independently varying firm-specific volatility, and GARCH behavior in the market-return shock In this framework, it can readily be shown that for portfolio, Pt; ðRDPt;t Þ2 Ds2b ðRM;t  Rf ;t Þ2 þ s2S;t ; ð1Þ P P % 2 ; s2S;t ¼ ½1=ðn  1Þ ni¼1 ðSi;t Þ2 ; bi is the marketwhere s2b ¼ ½1=ðn  1Þ ni¼1 ðbi  bÞ beta of firm i; b% is the mean market-beta across the firms, Si;t is the firm-specific component of firm i’s return, and n is the number of firms in the portfolio. s2S;t can be interpreted as an average residual variance from a market model estimation. The appendix provides details on the derivation of (1). In this simple framework, the estimated coefficient from regressing ðRM;t  Rf ;t Þ2 against ðRDPt;t Þ2 should be approximately equal to the cross-sectional variance of the firm’s market-betas. Further, standard GARCH should be adequate to model market volatility. RD should provide no incremental information about future market volatility because RD is simply a noisier measure of the period’s marketreturn shock, as shown in (1). 2.3. RD in a linear common-factor RGP with two (or more) common-factors, factor GARCH, constant factor-loadings, and independently varying firm-specific volatility In this simple two-factor framework, firm return dispersion is driven by the crosssectional dispersion in each vector of firm factor-loadings and idiosyncratic volatility. Then, the estimated coefficient from regressing ðRM;t  Rf ;t Þ2 against ðRDPt;t Þ2 would reflect a weighted average of the dispersion in each vector of factorloadings. Here, in contrast to Section 2.2, RD may provide incremental market-volatility information (information beyond that in market-return shocks). In this framework, GARCH behavior in market-level returns is due to volatility persistence in the two common-factor shocks. Then, if: (1) the two factors have different cross-sectional dispersion in their factor loadings, and (2) the two factors have

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different volatility persistence; RD should provide incremental market-volatility information. However, the sign of the relation between RD and future volatility is ambiguous. If the factor with higher volatility persistence has higher cross-sectional dispersion in its factor loadings, then we should observe a positive partial relation between RD and future market volatility. On the other hand, if the factor with lower volatility persistence has higher cross-sectional dispersion in its factor loadings, then we should observe a negative partial relation between RD and future market volatility. Under this subsection’s assumptions, if the true multiple common-factor RGP was known and the common-factor shocks were observable, then RD would provide no incremental market-volatility information beyond the information in the common-factor shocks. The part of RD that is attributable to dispersion in factorloadings could be directly calculated and the residual RD (RD that could not be attributed to dispersion in factor-loadings) would just reflect diversifiable risk. Further, if one directly controlled for the past shocks in each of the observable common factors, then the relation between RD and future market volatility should disappear. In real-world financial markets, though, the RGP is uncertain and common-factor shocks may be unobservable. Thus, in practice, RD may provide information about common-factor shocks and future market volatility, even if one directly controlled for shocks in observable common-factor proxies.

2.4. RD in a linear common-factor RGP with a common co-movement between market-level volatility, firm-specific volatility, and the dispersion in factor-loadings This framework allows for time variation in the dispersion of factor-loadings, which requires the addition of a t subscript to the factor-loading dispersion term in (1), ðs2b;t Þ: Here, high RD in a specific period may be attributed to the any of the following three items: (1) large common-factor shocks, (2) a high average firmspecific volatility across firms, and/or (3) a high cross-sectional dispersion in the factor-loadings. Or, perhaps more likely, high RD may be attributed to some combination of these three items. A common co-movement between market-level volatility, firm-specific volatility, and the dispersion in factor-loadings seems plausible, especially if time-varying RD has economic implications. In a common co-movement scenario, there are two direct empirical implications related to RD. First, in a regression of ðRM;t  Rf ;t Þ2 on ðRDPt;t Þ2 ; the estimated relation may be much greater than that implied by the dispersion in the unconditional market-betas across firms (because firm-specific volatility and/or the dispersion in factor-loadings are likely to be high when the absolute market return is large). Second, we may observe a positive partial relation between RD and future market volatility if the common co-movement behavior has intertemporal implications.

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3. Data description and analysis 3.1. Data description, U.S. monthly returns Our empirical analysis focuses on two U.S. equity return samples from CRSP. The first sample is the set of monthly returns of NYSE/AMEX stocks from July 1962 through December 1995. For this sample, we construct size-based, decile-portfolios from the individual firms, based on the equity market capitalization of each firm. The second sample is the set of monthly returns of NYSE stocks from January 1927 through June 1962. For this sample, we construct size-based, quintile-portfolios. We use quintile-portfolios for this earlier period so the number of stocks per portfolio will be similar to the number of stocks per decile-portfolio in the more recent data. For each month, all firms with return and market capitalization data that month are included when forming the portfolios. The portfolio returns are equally-weighted returns. We also use the individual firm returns to calculate a return dispersion (RD) measure for each portfolio, as discussed in Section 3.2. The number of firms in our large-firm portfolios ranges from 140 to 310 for the 1962–1995 NYSE/AMEX sample and from 88 to 215 for the 1927–1962 NYSE sample. In our empirical exploration, we focus on the return and RD of the largest decile (or quintile) portfolio, rather than using all firm returns, for the following reasons. First, these large-firm portfolios are good proxies for the overall value-weighted stock market. The correlation between the monthly NYSE/AMEX/NASDAQ valueweighted return from CRSP and our large-firm portfolio is 0.987 (0.990) for the 1962–1995 sample (1927–1962 sample). Second, a RD statistic computed from all firms may be overly influenced by small firms, due to high firm-specific volatility or non-synchronous trading. Third, large-firm returns have been shown to lead smallfirm returns (Lo and MacKinlay, 1990), which could cloud a RD statistic constructed from all firms. Finally, the use of size-based portfolios and their respective RD’s allows us to analyze both large and small-cap portfolios. 3.2. Firm return dispersion (RD) in U.S. monthly stock returns Firm return dispersion (RD) for portfolio Pt over period t is defined as follows: "

RDPt;t

n 1 X ¼ ðRi;t  RPt;t Þ2 n  1 i¼1

#1=2 ;

ð2Þ

where n is the number of firms in the portfolio, and subscript Pt indicates portfolio values. Descriptive statistics for the U.S. large-firm RD are provided in Table 1 for the two main samples. The means, medians, and standard deviations are comparable over subperiods except for the relatively large values in the pre-war depression subperiod. Fig. 1 plots the time series of large-firm RD’s and illustrates the variability of RD.

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Table 1 U.S. large-firm portfolios: portfolio returns and firm return dispersion (RD) across the portfolio’s component firms This table reports summary statistics for the portfolio returns and return dispersions of U.S. large-firm portfolios. For a given period t; a portfolio’s RDt is the cross-sectional standard deviation of the individual firm returns that comprise the portfolio. Panel A reports on monthly returns and RD’s from the largest size-based, decile-portfolio of NYSE/AMEX stocks from July 1962 through December 1995. Panel B reports on monthly returns and RD’s from the largest size-based, quintile-portfolio of NYSE stocks from January 1927 through June 1962. Sample period

Mean Median Std. (%) (%) Dev. (%)

Panel A: 7/62–12/95 sample Large-firm portfolio returns Overall 7/62–12/95 0.989 1st half 7/62–3/79 0.658 2nd half 4/79–12/95 1.32

1.13 0.730 1.50

4.27 4.29 4.23

Return dispersions of large-firm returns Overall 7/62–12/95 6.23 6.07 1.50 1st half 7/62–3/79 5.94 5.57 1.60 2nd half 4/79–12/95 6.51 6.16 1.34 Panel B: 1/27–6/62 sample Large-firm portfolio returns Overall 1/27–6/62 1.40 1st half 1/27–9/44 1.39 2nd half 10/44–6/62 1.40

1.54 1.39 1.71

7.03 9.23 3.75

Return dispersions of large-firm returns Overall 1/27–6/62 6.64 5.78 3.35 1st half 1/27–9/44 8.06 7.24 4.16 2nd half 10/44–6/62 5.22 5.04 1.08

Min. (%)

 21.3  11.4  21.3

10 25 75 90 Max. pct (%) pct (%) pct (%) pct (%) (%)

 4.11  4.76  3.42

 1.51  1.81  1.11

3.53 3.30 3.76

7.03 6.81 6.48

17.8 17.8 13.6

4.64 4.30 5.03

5.15 4.86 5.53

7.04 6.59 7.28

8.10 7.58 8.25

14.6 14.6 12.4

 28.4  5.62  28.4  8.38  9.90  3.90

 1.96  3.01  1.37

4.62 5.69 4.09

7.26 8.74 5.68

47.6 47.6 10.2

4.79 5.77 4.44

7.44 9.21 5.80

9.83 11.9 6.66

35.8 35.8 9.69

3.31 3.31 4.21

3.00 3.04 3.00

4.13 4.63 4.05

A simple autoregressive analysis of RD indicates that the variable is highly autocorrelated, but it is stationary. For the July 1962 to 1995 sample, the RD’s simple autocorrelation is 0.523 and the sum of the partial AR(1) through AR(5) coefficients is 0.771. For the earlier 1927 to June 1962 sample, the RD’s simple autocorrelation is 0.637 and the sum of the partial AR(1) through AR(5) coefficients is 0.812. We report five partial autocorrelations because the highest lag to have a significant partial autocorrelation is the fifth lag for either sample. 3.3. Daily U.S. equity returns We also construct an estimate of monthly stock market volatility from the daily returns within that month. For the July 1962 to 1995 period, we use the daily return series of the S&P 500 index from CRSP. For the earlier sample, we use daily U.S. stock market returns as in the Journal of Business, Schwert (1990).

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18

40

16 35

14 12

30

10 8

25

6 4

20

2 15

0

10 5

1993

1990

1987

1984

1981

1978

1975

1972

1969

1966

1963

1960

1957

1954

1951

1948

1945

1942

1939

1936

1933

1930

1927

0

Fig. 1. Monthly time-series of U.S. large-firm portfolio returns and firm return dispersions, in %. This figure plots the monthly time-series of: (1) absolute large-firm portfolio returns (bottom series, left axis); and (2) large-firm return dispersions (top series, right axis). The portfolio return and firm return dispersions are formed from the largest size-based decile of NYSE/AMEX stocks for the more recent July 1962 through 1995 period, and from the largest size-based quintile of NYSE stocks for the earlier 1927 through June 1962 period.

3.4. Other U.S. data From the National Bureau of Economic Research (NBER), we obtain the official business cycle data. From the Federal Reserve Statistical Release, we obtain: (1) bond default yield spread data, as defined by the spread between the Moody’s Baa bond yield minus the Aaa bond yield; and (2) 3-month constant maturity T-bill rates, for use as a risk-free rate. For U.S. government bond returns, we calculate returns from the Fama-Bliss 5-year government bond prices from CRSP. Finally, we thank Eugene Fama for providing the monthly return data for the three factor-mimicking portfolios from Fama and French (1993).

3.5. International stock market returns We obtain stock returns for Japan and the United Kingdom from Datastream International to explore the generality of our findings. We collect monthly returns for the firms that comprise the Nikkei-225 for Japan and the FTSE-100 for the U.K. over the 1980 through 1999 period. The firm return dispersion for each market is calculated from the available firm returns in Datastream.

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4. Main empirical results: firm return dispersion and future market volatility 4.1. Main results We next turn to our main empirical investigation. We examine the relation between firm return dispersion and future market-level volatility in U.S. stock returns over the 1927–1995 period. For our primary empirical model, we use an extended version of the asymmetric GARCH(1,1) model of Glosten et al. (1993) (GJR). We make this choice for several reasons. First, a GARCH framework enables us to estimate the effects of lagged RD on the conditional mean and conditional volatility jointly. Second, Engle and Ng (1993) find that the GJR asymmetric specification is among the best parametric GARCH specifications in a horse race between models. Finally, a parsimonious widely-used GARCH(1,1) specification is desirable because of the limited number of observations with monthly data, because it adds objectivity to our testing, and because it facilitates comparison of our results to prior work. As reported later, it is important to note that we have estimated other ARCH/GARCH specifications and find similar results. Our primary GARCH specification is as follows: mean: variance:

Ret ¼ b0 þ b1 Ret1 þ b2 RDt1 þ et ;

ð3Þ

2 Vt ¼ d0 þ d1 e2t1 þ d2 D t1 et1 þ d3 Vt1 2 þ d4 RD2t1 þ d5 D t1 RDt1 ;

ð4Þ

where Ret is the excess return of our large-firm portfolio; RDt is the return dispersion for our large-firm portfolio; Vt is the conditional variance; D t1 is a dummy variable that equals one if the lagged return residual ðet1 Þ is negative and is 0 otherwise; and the b’s and d’s are estimated coefficients. Note that this specification symmetrically allows both e2t1 and RD2t1 to have a different association with future volatility when et1 is negative.5 In the estimation, a log likelihood function is maximized, assuming conditional normality of the market-return shock et : The t-statistics reported in the tables are asymptotic t-statistics, calculated with standard errors that assume conditional normality. Bollerslev and Wooldridge (1992) derive a method to calculate standard errors that are robust to departures from the conditional normality assumption. We also estimate and report Bollerslev/Wooldridge robust standard errors for our main results.6 The GARCH results are presented in Table 2. Panel A presents the 7/62 to 12/95 sample and Panel B presents the older 1/27 to 6/62 sample. Rows 1 and 3, 5 Allowing for this sign asymmetry for the RD term is also suggested by Lamoureux and Panikkath (1994), who find that cross-sectional dispersion is larger when the market moves up than when it moves down. 6 The use of large-firm monthly portfolio returns, which are fairly close to normally distributed, suggest that we will find only minor differences for the Bollerslev/Wooldridge standard errors.

Ret ¼ b0 þ b1 Ret1 þ b2 RDt1 þ et ; variance :

2 2  2 Vt ¼ d0 þ d1 e2t1 þ d2 D t1 et1 þ d3 Vt1 þ d4 RDt1 þ d5 Dt1 RDt1 ;

6.

5.

Unrestricted (1st half) Unrestricted (2nd half)

Panel A: U.S. monthly Model 1. Restrict d2 ¼ d4 ¼ d5 ¼ 0 2. Restrict d2 ¼ d5 ¼ 0 3. Restrict d4 ¼ d5 ¼ 0 4. Unrestricted

e2t1

7/62–12/95 n ¼ 402 7/62–12/95 n ¼ 402 7/62–12/95 n ¼ 402 7/62–12/95 n ¼ 402 7/62–3/79 n ¼ 201 4/79–12/95 n ¼ 201

0.096 (2.98) 0.050 (1.49) 0.035 (1.35)  0.075 ( 5.82)  0.069 ( 3.91)  0.033 ( 0.61)

equity returns, 7/62–12/95

Sample

d1

0.135 (2.61) 0.058 (0.57) 0.138 (0.86) 0.069 (0.24)

d2 2 D t1 et1

0.874 (23.1) 0.839 (15.8) 0.872 (21.5) 0.120 (1.09) 0.392 (1.67) 0.164 (1.05)

d3 Vt1

0.154 (3.42) 0.087 (1.28) 0.014 (0.13)

0.062 (2.05)

d4 RD2t1

0.384 (5.56) 0.301 (2.67) 0.397 (3.87)

d5 2 D t1 RDt1

8.80 [0.012] 35.68 [0.000] 17.04 [0.000] 0.38 [0.825]

Wald test, d1 ¼ d2 ¼ 0 [Chi-sq. (2)]

50.48 [0.000] 8.21 [0.016] 16.16 [0.000]

Wald test, d4 ¼ d5 ¼ 0 [Chi-sq. (2)]

0.193

0.245

0.222

0.222

Std. Dev. of RD2t1 ð100Þ

0.422

0.350

0.382

0.382

Std. Dev. of e2t1 ð100Þ

1103.0

1079.9

1080.7

1075.9

Log likelihood

where Ret is the large-firm portfolio excess return (return less risk-free rate), RDt1 is the lagged large-firm return dispersion, Vt is the conditional variance, D t1 is a dummy variable that equals one if the lagged return residual (et1 ) is negative and is 0 otherwise, and the b’s and d’s are estimated coefficients. The system is estimated simultaneously by maximum likelihood estimation using the conditional normal density. Panel A provides results for our U.S. large-firm portfolio of the largest decile of NYSE/AMEX stocks from July 1962 through December 1995. Panel B provides results for our earlier U.S. large-firm portfolio of the largest quintile of NYSE stocks from January 1927 through June 1962. T-Statistics are in parentheses. For the Wald tests on joint restrictions, the Chisquared p-value is in brackets.

mean :

Table 2 Main intertemporal results: firm return dispersion and future stock market volatility The table reports results from estimating the following GARCH system on U.S. monthly stock returns:

398 C.T. Stivers / Journal of Financial Markets 6 (2003) 389–411

6.

5.

Unrestricted (1st half) Unrestricted (2nd half)

Panel B: U.S. monthly Model 1. Restrict d2 ¼ d4 ¼ d5 ¼ 0 2. Restrict d2 ¼ d5 ¼ 0 3. Restrict d4 ¼ d5 ¼ 0 4. Unrestricted

1/27–6/62 n ¼ 426 1/27–6/62 n ¼ 426 1/27–6/62 n ¼ 426 1/27–6/62 n ¼ 426 1/27–11/41 n ¼ 179 1/46–6/62 n ¼ 198

0.163 (5.15) 0.120 (3.49) 0.051 (1.17) 0.0051 (0.11) 0.017 (0.21)  0.042 ( 0.51)

equity returns, 1/27–6/62

0.172 (2.60) 0.137 (1.84) 0.073 (0.63) 0.152 (1.08)

0.829 (30.6) 0.820 (25.8) 0.832 (30.6) 0.828 (23.7) 0.805 (11.0) 0.701 (6.54) 0.050 (1.67)  0.011 ( 0.23) 0.117 (1.44)

0.070 (2.20)

0.072 (1.54) 0.272 (2.20) 0.192 (1.67)

26.22 [0.000] 6.52 [0.038] 1.04 [0.593] 1.16 [0.557] 8.52 [0.014] 4.86 [0.088] 4.76 [0.093] 0.116

1.45

1.00

1.00

0.191

2.26

1.58

1.61

1046.8

1039.5

1040.0

1036.4

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respectively, report the results from estimating a standard GARCH(1,1) model and a GJR asymmetric GARCH model. The results indicate that the well-known ARCH behavior is evident in the market returns of both samples. Row 2 of Table 2 provides the results from estimating a standard GARCH(1,1) model that includes an RD explanatory term but restricts d2 ¼ d5 ¼ 0 in (4). For both samples, the estimated d4 coefficient on the RD term is sizeable, positive, and statistically significant. Also note that the estimated d1 coefficient on the e2t1 term decreases and is no longer significant in Panel A. Next, Row 4 of Table 2 provides the results from estimating the full GARCH system in (3) and (4). We find a reliable positive relation between the conditional variance and RD2t1 for both samples. Further, the results indicate that the wellknown positive relation between market volatility and the lagged market-return shock largely disappears when controlling for RD. Table 2 also reports a Chi-squared test statistic from a Wald test of the joint restriction that d4 and d5 of (4) are jointly zero. For both samples, the restriction is rejected with a maximum p-value of 0.014. We find that Bollerslev/Wooldridge standard errors for the d4 and d5 coefficients are similar at 3:01=3:21 for d4 =d5 for Panel A and 1:56=0:97 for d4 =d5 for Panel B. In terms of the skewness and excess kurtosis of the standardized return residuals, our GARCH model in (3) and (4) performs well as compared to the wellknown GARCH models. For Panel A, the standardized residual for the GJR model (Row 3) has a skewness of 0:468 (p-value of 0:0001) and an excess kurtosis of 2:354 (p-value of 0:0000). For our RD model (Row 4), the standardized residual has a skewness of 0:108 (p-value of 0:379) and an excess kurtosis of 0:487 (p-value of 0:0479). For the earlier sample in Panel B, the standardized residual for the GJR model (Row 3) has a skewness of 0:357 (p-value of 0:0027) and an excess kurtosis of 0:745 (p-value of 0:0020). For our RD model (Row 4), the standardized residual has a skewness of 0:235 (p-value of 0:048) and an excess kurtosis of 0:659 (p-value of 0:0059). Finally, we note that all of the estimated b coefficients in the conditional mean equation are insignificant for both samples. To save space, we do not report the estimated b coefficients in the table. To conclude, our GARCH model in (3) and (4) performs well in terms of the statistical significance of the coefficients, the increased log-likelihood function value, and the reduction in skewness and kurtosis of the standardized residual. It is a simple specification that treats the lagged returns shocks and lagged RD symmetrically. Additionally, we have performed Engle and Ng’s (1993) sign and size bias tests on this volatility model and found no evidence of model mis-specification.7 Accordingly, we use the GARCH model in (3) and (4) as the primary GARCH model in this 7

Specifically, we perform the ‘sign bias’, ‘positive size bias’, and ‘negative size bias’ specification tests of Engle and Ng on the conditional volatility model. These tests examine whether the squared normalized residual can be predicted by using other information from the past return shocks, such as the sign or size non-linearities. If so, then the tests indicate that the volatility model is mis-specified.

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paper. As reported in Section 4.3, our findings are robust to other alternate GARCH specifications. Next we estimate the GARCH system of (3) and (4) over subsamples covering 1=27 to 11=41; 1=46 to 6=62; 7=62 to 3=79; and 4=79 to 12=95: (We omit the World War II years during the subperiod analysis.) The results are presented in Table 2, Rows 5 and 6. The results are consistent with the full sample results. For these subperiods, the positive relation between RD and future market volatility remains evident and is statistically significant. In contrast, there is no reliable positive relation between the conditional variance and the lagged market-return shock for any of these subperiods. 4.2. Impulse response curves implied by the GARCH estimation To further illustrate the sensitivity of the conditional market variance to RDt1 and the market return residual ðe2t1 Þ; we present simple impulse response curves in Fig. 2 for the 7=62 to 12=95 sample. Panel A presents the impulse response curve for the conditional market variance as a function of RDt1 for the system of (3) and (4). Panel B presents the impulse response curve for the conditional market variance as a function of e2t1 for the GJR model.8 Our procedure follows from Engle and Ng (1993). These curves show how the conditional volatility varies with the respective explanatory variable, with all other explanatory variables set at their unconditional expected values. Each point on these curves corresponds to an actual observation for the respective explanatory variable. These curves illustrate both the substantial relation between RD and future market volatility and the sign asymmetry, where RD during market down movements has a relatively higher association with future market volatility. 4.3. Other robustness checks We estimate alternate ARCH/GARCH models that each include the lagged RD as an explanatory variable in both the conditional mean and variance equations. First, we consider non-linearities by estimating a GARCH specification motivated by Hentschel (1995). Specifically, we replace the squared return shocks in the conditional variance equation with the absolute return shock raised to the threehalves power. Next, we estimate a simpler ARCH model that includes the RD terms. Finally, we extend the model to include a GARCH-in-mean term in the conditional mean equation. For each model, the intertemporal volatility information in RD remains reliably evident. 8 We show only the 7=62 to 12=95 sample for brevity and because these curves are more likely to reflect current market relations. Impulse response curves for the older sample are similar to Panel B in Fig. 2 for both et1 and RDt1 : We use the GJR model to illustrate a system that does not include RD and because, in our primary RD model, the incremental relation of the conditional variance to e2t1 is small and marginally negative.

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Contribution of RD t −1 to the Conditional Variance (Primary RD Model)

Variance (in monthy % units)

60 50 40 30 20 10 0 -15

-10

-5

0

5

10

15

RD(t-1)*(Sign of Market-Return Residual at t-1) in %

(A)

Contribution of Market-return Residual ( ε t2−1 ) to the Conditional Variance (GJR model)

Variance (in monthly % units)

60 50 40 30 20 10 0 -20

(B)

-16

-12

-8

-4

0

4

8

12

16

20

Market-Return Residual at (t-1) in %

Fig. 2. Impulse response curves for the conditional market variance. This figure presents impulse response curves for the monthly conditional market variance as a function of RD (for the system of (3) and (4)(A)) and the market-return residual (for the GJR model (B)). The sample period is 7/62 to 12/95. These curves show how the conditional variance varies with the respective variable, with all other explanatory variables set at their unconditional expected values. Each point on the curves corresponds to an actual observation for the respective explanatory variable. Panel A: Contribution of RDt1 to the conditional variance (Primary RD Model); Panel B: Contribution of Market-return Residual ðe2t1 Þ to the conditional variance (GJR model).

Next, we estimate our primary GARCH model in (3) and (4) on the portfolio returns and RD’s for the smaller size-based portfolios. For this estimation, the RD is the cross-sectional return dispersion for the firms that comprise each respective small-firm portfolio. We estimate the model on deciles 1 through 9 for the more

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current sample and quintiles 1 through 4 for the older sample. We find that a portfolio’s RD is positively related to the future volatility of the portfolio (at a 5% pvalue, or better) for all the smaller-firm portfolios except one. For the second smallest quintile-portfolio in the older sample, this relation is also positive but the p-value is only 0:069: Finally, we analyze the intertemporal relation between market volatility and the lagged RD with an alternate measure of market volatility. We form a monthly volatility estimate from the daily returns within that month, along the lines of French et al. (1987) and Schwert (1989). We then estimate vector autoregressive models in volatility and RD with this volatility estimate. We find the same positive, partial relation between RD and future market volatility with this alternate approach. 4.4. Out-of-sample investigation Next, we perform an out-of-sample exercise. We compare our primary GARCH model in Eqs. (3) and (4) to the GJR GARCH model for our 1962–1995 sample.9 We use a 201-month, first-stage estimation period and then calculate the one-step ahead conditional volatility, based on the parameters from the first-stage estimation period. We roll forward through the sample to obtain 201 out-of-sample conditional volatility estimates. We use a substantial estimation period of 201 months because the GARCH volatility equation has five coefficients to estimate. In this out-of-sample exercise, we find that our RD model outperforms the GJR GARCH model. In an OLS regression of the out-of-sample forecasts against the realized volatility, the estimated coefficient on the forecast from our RD model is strongly positive with a t-statistic of 3:09: In contrast, the estimated coefficient on the forecast from the GJR-GARCH model is marginally positive with a t-statistic of 1.87.10 4.5. International evidence Finally, we present international evidence by investigating the aggregate stock market returns and firm return dispersions of the Japanese and U.K. equity markets. The samples have 240 monthly return observations, spanning from 1980 through 1999. We estimate the GARCH system in (3) and (4) for the two countries. We find that the lagged RD terms are jointly significant in the conditional variance equation for 9

We focus on the more current sample because the earlier sample has several very distinct periods, (the depression, World War II, and the 1950s) which seem likely to have substantially different parameters. 10 The correlation between these two out-of-sample conditional volatility series is 0:41: This modest correlation is because variation in the conditional volatility from our full model is largely driven by recent RD, while variation in the conditional volatility from the GJR model is driven solely by recent marketreturn shocks.

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both the U.K. (p-value of 0:025) and Japan (p-value of 0:094). We conclude that the results for the non-U.S. markets are qualitatively similar to the results for the U.S. market.

5. Additional evidence: controlling for economic conditions and other market variables 5.1. Economic explanatory variables We examine whether the volatility information in RD is robust to controlling for three different economic variables suggested by prior literature. We add these additional variables to both the mean and variance equation of our primary GARCH model, (3) and (4), and re-estimate the system. Due to data availability, we only evaluate the more recent sample. First, we consider the risk-free rate from T-bills, which is motivated by Glosten et al. (1993). They find that the risk-free rate is an important factor in the variance equation and that standard GARCH relations in the variance equation decrease dramatically with the inclusion of the risk-free rate. Second, we consider the bond yield spread between Moody’s Baa and Aaa bonds, which is motivated by Schwert (1989). He find that the bond yield spread is one of the stronger variables related to time-varying stock volatility. Third, we evaluate dummy variables that control for whether the economy is in a recessionary or expansionary state as defined by the NBER, which is motivated by Schwert (1989). Schwert finds that volatility is reliably higher during recessions. We estimate the following GARCH system for each of the economic variables: mean : variance :

Ret ¼ b0 þ b1 Ret1 þ b2 RDt1 þ b3 EVt1 þ et ;

ð5Þ

2 2 Vt ¼ d0 þ d1 e2t1 þ d2 D t1 et1 þ d3 Vt1 þ d4 RDt1 2 þ d5 D t1 RDt1 þ d6 EVt1 ;

ð6Þ

where EV is one of the three variables discussed in the preceding paragraph. The other variables are as defined for (3) and (4). The results are reported in Table 3. We find the following. First, RD remains important in explaining future stock market volatility. See models 3-1B, 3-2B, and 33B. The final row of the table reports that a test of the joint restriction that d4 ¼ d5 ¼ 0 is rejected, with a p-value o0:001 for each model. Second, we find that only the bond yield spread remains an important explanatory variable for volatility when the RD terms are included. Third, we note that the d3 coefficient on Vt1 decreases dramatically for all of the extended models, as compared to the d3 coefficient for the standard GARCH model in Table 2. For models 3-1B, 3-2B, and 3-3B, the estimated d3 coefficient is less than 0:1: This suggests that firm RD and economic variables largely subsume the volatility information in variables from period t  2 and older. The estimated b1 ; b2 ; and b3 ’s are statistically insignificant.

Ret ¼ b0 þ b1 Ret1 þb2 RDt1 þ b3 EVt1 þ et ; variance:

2 2  2 Vt ¼d0 þd1 e2t1 þd2 D t1 et1 þ d3 Vt1 þ d4 RDt1 þ d5 Dt1 RDt1 þ d6 EVt1 ;

0.210 (3.44) 1089.1

 0.070 ( 4.91) 0.300 (2.24) 0.391 (2.23)

3-1A

 0.061 ( 2.59) 0.025 (0.21) 0.099 (0.89) 0.106 (1.35) 0.454 (5.37) 0.038 (0.74) 1101.1 33.9 [0.000]

3-1B

EV ¼ risk-free rate

0.0010 (4.28) 1095.1

 0.100 ( 8.71) 0.425 (3.42) 0.350 (2.26)

3-2A  0.059 ( 5.52) 0.035 (0.38) 0.012 (0.09) 0.048 (0.69) 0.395 (5.22) 0.0011 (3.11) 1115.5 28.6 [0.000]

3-2B

EV ¼ bond yield spread

0.0012 (1.72) 1081.4

 0.070 ( 2.84) 0.281 (2.14) 0.174 (0.52)

3-3A  0.068 ( 6.02) 0.016 (0.15) 0.027 (0.32) 0.150 (3.90) 0.488 (5.74) 0.0004 (0.66) 1104.7 59.0 [0.000]

3-3B

EV ¼ recession state

0.049 (1.04) 0.194 (2.60) 0.701 (9.68)

SMB :3 0:18 HML :3 0:09 1085.6

3-4A

3-5A

 0.072 ( 8.11) 0.064 (0.60)  0.036 ( 0.34) 0.132 (3.65) 0.354 (4.17) 0.322 (1.66) 1101.7 45.5 [0.000]

3-5B

EV ¼ bond returns

 0.074  0.045 ( 3.63) ( 1.00) 0.070 0.307 (0.72) (2.56) 0.196 0.552 (1.77) (4.94) 0.173 (2.44) 0.382 (4.99) 0:113 0.534 0:083 (3.82) 1104.4 1087.5 45.8 [0.000]

3-4B

EV ¼ SMB; HML

Notes: 1. n ¼ 402 observations; 2. only the coefficients for the variance equation are shown for brevity; 3. these cells report the two coefficients that were estimated on the lagged SMB and HML squared returns in the conditional variance equation, respectively.

Likelihood value Chi-sq. Wald test, Joint d4 ¼ d5 ¼ 0

d6 ðEVt1 Þ

2 d5 ðD t1 RDt1 Þ

d4 ðRD2t1 Þ

d3 ðVt1 Þ

2 d2 ðD t1 et1 Þ

d1 ðe2t1 Þ

returns, 7/62–12=951;2

U.S. monthly

where EVt equals the following, for: (3-1) the risk-free rate from T-bills; (3-2) the yield spread between Moody’s Baa and Aaa corporate bonds; (3-3) one if the economy was in a recession that period, and zero otherwise; (3-4) both the Fama-French SMB and HML portfolios returns in the mean equation and the SMB and HML squared-portfolio returns in the variance equation; and (3-5) the bond return for 5-year government bonds in the mean equation and the squared bond return in the variance equation. The sample period is 7/62–12/95. The other terms are as defined in Table 2. T-Statistics are in parentheses. For the Waldhypothesis tests, the p-value of the Chi-squared statistic is in brackets.

mean:

Table 3 Firm return dispersion and future stock market volatility: while controlling for economic conditions and other market variables This table presents the results from estimating the following GARCH system on our sample of monthly U.S. large-firm portfolio returns:

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We conclude that the positive relation between RD and future market volatility remains strong when controlling for these additional economic variables.11 Further, these results seem consistent with the limited success of prior studies that have attempted to explain time-varying stock volatility with other economic and market variables, see Schwert (1989). 5.2. Multiple common factors Next, we examine whether our RD findings are robust to controlling for some widely-used common-factor proxies. As discussed in Section 2.3, if the volatility information in RD is because RD is capturing information about multiple commonfactor shocks, then the importance of RD should diminish if one directly controls for the prior common-factor shocks. We estimate a variation of the models in (5) and (6), where the returns of the common-factor proxies are explanatory terms. First, we control for the lagged return-shocks of the three factor-mimicking portfolios from Fama and French (1993). We add the SMB and HML portfolio returns as explanatory terms in the model. Second, we control for the lagged return-shocks of government bond returns. The use of bond returns is motivated by Scruggs (1998) and Schwert (1989). Scruggs considers a two-factor model motivated by Merton’s intertemporal CAPM, in which the bond return proxies for the second hedge factor. Schwert finds evidence of a modest relation between bond return volatility and equity volatility. The results are presented in Table 3, models 3-4 and 3-5. We find that controlling for these proposed factors does not diminish the volatility information in RD. Further, neither the bond returns nor the factor-mimicking portfolios reliably improve the model for market-level volatility. While this evidence is not supportive of an omitted factor explanation for the volatility information in RD, it is far from conclusive. Identifying variables that may proxy for omitted common-factors is tenuous and future research in this area might prove interesting.

6. The contemporaneous relation between RD and the market-return shock The discussion in Section 2 also suggests implications for the contemporaneous relation between firm RD and the market-return shock. As suggested by (1), we estimate the following model on our sample of monthly U.S. stock returns: ðRDPt;t Þ2 ¼ g0 þ g1 ðRMkt;t  Rf ;t Þ2 þ et ;

ð7Þ

where RDPt;t is the cross-sectional standard deviation of the individual firm returns that comprise portfolio Pt in period t; RMkt;t is the return of our large-firm market 11 Motivated by GJR (1993), we also re-estimate the GARCH system (5) and (6) where EV ¼ January and October dummy variables in the mean and variance equation. Our RD findings are robust to these seasonals.

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portfolio; Rf ;t is the risk-free rate from T-bills; et is the residual, and the g’s are estimated coefficients. We estimate (7) for two different portfolios for each of our main sample periods (7/62–12/95 and 1/27–6/62). First, we evaluate our large-firm portfolios, which are comprised of the top decile (or quintile) of large firms based on the current market capitalization. Second, we evaluate a constant-firm portfolio for each sample period, which have a fixed set of component firms. For the constant-firm portfolios, we select the NYSE firms that have complete monthly return data over the respective sample period (n ¼ 324 firms for the more recent sample and n ¼ 239 firms for the older sample). For the constant-firm portfolios, we calculate a market-model beta for each individual firm in each portfolio over the entire sample period; and then we compute the actual cross-sectional dispersion in the calculated market-betas ðs2b Þ for the portfolio. We can then compare this sample s2b to the g1 coefficient obtained from estimating (7). The results are reported in Table 4. We find that the estimated g1 ’s are much larger than the sample s2b : For the constant-firm portfolios, the estimated g1 ’s are 0:303 (newer sample) and 0:710 (older sample) versus a sample s2b of 0:089 and 0:166 for

Table 4 The contemporaneous relation between firm return dispersion (RD) and the squared market return This table reports the results from estimating the following model on our sample of monthly U.S. largefirm returns: ðRDPt;t Þ2 ¼ g0 þ g1 ðRMkt;t  Rf ;t Þ2 þ et ; where RDPt;t is the cross-sectional standard deviation of the individual firms that comprise the portfolio Pt in period t; RMkt;t is the return of our large-firm market portfolio; Rf ;t is the risk-free rate from T-bills; et is the residual, and the g’s are estimated coefficients. T-Statistics are in parentheses, calculated with heteroskedastic and autocorrelation consistent standard errors by the Newey and West (1987) method. Rows 1 and 2 report on the 7/62–12/95 period where the large-firm portfolio is the largest size-based, decile-portfolio of NYSE/AMEX stocks. Rows 3 and 4 report on the 1/27–6/62 period where the largefirm portfolio is the largest size-based, quintile-portfolio of NYSE stocks. For rows 2 and 4, the constantfirm portfolios are comprised only of NYSE stocks that have monthly returns over the entire period, n ¼ 324 firms for the more recent period and n ¼ 239 firms for the earlier period.

1. 2. 3. 4. a

Portfolio Pt (RD source)

Sample period

Large-firm portfolio Constant-firm portfolio Large-firm portfolio Constant-firm portfolio

7/62–12/95 ðn ¼ 402Þ 7/62–12/95 ðn ¼ 402Þ 1/27–6/62 ðn ¼ 426Þ 1/27–6/62 ðn ¼ 426Þ

g0 ð100Þ 0.359 (19.7) 0.480 (27.3) 0.310 (11.7) 0.623 (10.3)

g1 0.273 (2.88) 0.303 (4.12) 0.475 (8.53) 0.710 (12.4)

R-squared (%)

s2b a

21.7

n/a

21.6

0.089

80.8

n/a

52.2

0.166

For the constant-firm portfolios, this column reports a direct estimate of the cross-sectional variance of the individual firm market-betas. For this direct estimate, we use unconditional market-betas that are calculated from a market-model estimation on each of the individual firms over the respective sample period.

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the respective samples. The estimated g1 ’s for our large-firm portfolios are also sizable and seem too large to reflect plausible values of the dispersion in unconditional market-betas. From the perspective of Section 2.4, these findings suggest that firm-specific volatility and/or the dispersion in factor loadings vary positively with the magnitude of the market return.

7. Conclusions and discussion of results The primary implication of our work is that return dispersion across firms (which, at first glance, may be discounted as only reflecting idiosyncratic volatility) seems very relevant in incrementally explaining future market-level volatility. Our results indicate that simple return-dispersion metrics parsimoniously capture information about the market volatility environment. We study U.S. monthly stock returns over the 1927–1995 period and find a sizable and reliable positive relation between firm return dispersion and future market-level volatility. Further, the positive relation between market-return shocks and future market-level volatility (the well-known GARCH relation) largely disappears when controlling for firm return dispersion. Additionally, we find that the standard GARCH relation in market returns is much weaker when controlling for the risk-free interest rate, the bond default yield spread, and the recessionary state. In contrast, the positive relation between RD and future market volatility remains strong when controlling for these economic variables. Our RD findings are also robust to controlling for prior return-shocks in Treasury bonds and in the three factor-mimicking portfolios of Fama and French (1993). Finally, we find a substantial positive contemporaneous relation between firm RD and the squared excess market return. This relation is greater than the relation predicted by a market-model process with constant market-betas and firm-specific volatility that only varies randomly. Collectively, our contemporaneous and intertemporal findings suggest that there is a common co-movement between market-level volatility, firm-specific volatility, and the dispersion in factor loadings; and that these interrelations mean that RD captures incremental information about future market volatility. Under the assumptions of a known linear, common-factor, return-generating process with observable common-factor shocks, RD should only provide information about diversifiable risk. However, we find that time-varying RD has future market-wide implications, even when controlling for various common-factor proxies. In the sense of Cochrane (2001), this suggests that RD may serve as a state variable in markets with uncertain return-generating processes and/or unobserved common-factor shocks.12 12 Cochrane (2001) notes that state variables should ‘‘describe the conditional distribution of asset returns the agent will face in the future’’; and, ‘‘that in multiple goods models, relative price changes are also state variables’’.

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Here, we briefly discuss related evidence on economic interpretation. First, Christie and Huang (1994a) find that firm return dispersion is systematically higher during recessions and when the bond default yield spread is high. These patterns are also evident in our sample. For the July 1962 to 1995 sample, the simple correlation between RD and the bond yield spread is 0.44. The monthly firm RD in recessions is greater than the sample’s median RD for 87.7% of the recessionary periods. Second, Loungani et al. (1990) find that RD leads unemployment. Third, RD is positively correlated with aggregate firm-level volume (Bessembinder et al., 1996); and high volume may reflect economic portfolio reallocations with a changing investment opportunity set (Lo and Wang, 2000). Next, technical indicators of divergent returns have been interpreted as reflecting uncertain stock markets (Fosback, 1993). An example of a specific theoretical framework where RD might provide incremental economic information is Veronesi (1999). In his model, there is parameter uncertainty about the economic state. Price movements depend upon both the period’s information signal and the relative uncertainty about the economic state. Times with higher economic uncertainty are associated with higher price responsiveness to news, which might generate high RD and higher market-level volatility. However, his model is a single risky asset model. It would be interesting if extensions of his model (or others) to multiple risky assets might generate our findings as an empirical implication. Our findings have implications for future research. First, as proposed in the preceding paragraph, our evidence suggests direction for future work on volatility and asset pricing. Second, work to refine the market-wide economic implications of time-varying RD might prove interesting. Finally, are there other applications where a RD metric may prove useful?

Appendix. derivation of Eq. (1) This appendix provides details for the derivation of Eq. (1), which is the case of firm return dispersion (RD) in a single, linear market-factor return generating process with constant market-betas. For this simple benchmark, we consider the case with constant firm betas and independently-varying firm-specific volatility. Since our focus is on volatility and a firm’s short-horizon expected return is small as compared to its volatility, we use the expected return expression from CAPM as an approximation, or EðRi;t Þ ¼ Rf ;t þ bi ½EðRM;t Þ  Rf ;t ;

ðA:1Þ

where ½EðRM;t Þ  Rf ;t  is the market risk premium for the single market factor, Rf ;t is the risk free rate, RM;t is the market return, and bi is the factor loading for security i: Then the return generating process for firm i in period t is Ri;t ¼ Rf ;t ð1  bi Þ þ bi RM;t þ Si;t ;

ðA:2Þ

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where Si;t is the firm-specific component of a firm’s return. The Si;t ’s are uncorrelated with the market return (by definition), and are assumed to be uncorrelated with a firm’s bi : Next, substitute (A.2) into (2) of the main text, and square both sides of the equation: n 1 X ðRDPt;t Þ2 ¼ ½Rf ;t ð1  bi Þ þ bi RM;t þ Si;t  RPt;t 2 : ðA:3Þ n  1 i¼1 Note that the portfolio return is approximately equal to the following expression of the market return: % M;t ; RPt;t D½Rf ;t ð1  mb Þ þ bR ðA:4Þ Pn where we express the mean b for the firms in the portfolio as b% ðb% ¼ ð1=nÞ i¼1 bi Þ: This expression is approximate, because it assumes that the firm-specific Si;t ’s perfectly sum to zero. In our empirical work, we focus on large, well-diversified portfolios that have correlations of nearly one with the value-weighted equity market. Thus, for the primary portfolios in our analysis, the Si;t ’s do approximately sum to zero, so (A.4) is a good approximation. Next, substitute out the portfolio return in (A.3) by substituting (A.4) into (A.3) to yield n 1 X % þ ðSi;t Þ2 : ðRDPt;t Þ2 D ½ðRM;t  Rf ;t Þðbi  bÞ ðA:5Þ n  1 i¼1 Then square the terms inside and carry out theP summation to obtain P the summation, % 2 and s2S;t ¼ ½1=ðn  1Þ ni¼1 ðSi;t Þ2 : Note that (1) where s2b ¼ ½1=ðn  1Þ ni¼1 ðbi  bÞ the cross-terms should sum to approximately zero since the Si;t ’s are uncorrelated with the market return and firm betas. Eq. (1) expresses the return dispersion as a function of the excess market return, the cross-sectional variation in market-betas, and the average idiosyncratic volatility.

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